In Denmark (if you guys are foodies), book in NOW to get a table (if you are lucky) at Noma (2 star Michelin restaurant) or Restaurant Geranium (3 star Michelin restaurant) Both restaurants serve "the new Scandinavian" cuisine.  If you take the trip to Oslo, Norway there is another 3 star Scandinavian Cuisine restaurant there - Maaemo - but be warned - these places are expensive, but will probably provide a culinary experience you have not had before. :)

Copenhagen's biggest attraction is probably Tivoli and the Little Mermaid, but Christiania might be interesting for its weirdness.. 

Amsterdam has the weirdest tourist attraction of all of the cities you have mentioned, if you ask me: The red light district. Seeing families with the children, mom, dad and the grandparents walk through the hooker district was a weird experience.. :D

BTW: My biggest tip for traveling these days is to use Tripadvisor to find good places to eat, has worked really well most anywhere we have been traveling! Having the app on your phone (given that you have gotten yourself a local SIM, so you don't go broke when you come home and get the phone bill) really is a great way to find a good place to eat when you find out you are about to starve to death :) Has made sure we have avoided the biggest turist traps and have found some real gems. :)